franco moriconi - actuaries.chb9f3124e...bf-type model by saluz, gisler and wuthric h (2011). saa...

Stochastic Claims Reserving

with Credible Priors

Franco Moriconi

University of Perugia

SAA Annual Meeting – Fribourg, september 2, 2016

The Claims Reserving Problem: standard setup

We have the rectangle of incremental claims Xi,j where:

· i = 0, 1, . . . , I is the index of the accident year (AY)

· j = 0, 1, . . . , J is the index of the development year (DY)

(the index t = i + j denotes the calendar year )

The cumulative claims of AY i up to DY j are defined as:

Ci,j =

j∑k=0

Xi,k ,

We assume that all claims are settled after development year J where J ≤ I .

Assume we are at time I and let us consider the sets:

DI = Xi,j; i + j ≤ I, j ≤ J, DcI = Xi,j; i + j > I, i ≤ I, j ≤ J.

=⇒· the random variables Xij ∈ DI are observed

· the random variables Xij ∈ DcI – the outstanding claims – must be predicted .

SAA Annual Meeting – Fribourg, september 2, 2016 2

X0,0 X0,1 · · · · · · · · X0,J−1 X0,J

X1,0 X1,1 · · · · · · · · X1,J−1 X1,J...

............

......

.........

......

XI−J,0 XI−J,1 · · · · · · · · XI−J,J−1 XI−J,JXI−J+1,0 XI−J+1,1 · · · · · · · · XI−J+1,J−1 XI−J+1,J

· · · · · · · · · · · ·...

............

......

.........

......

· · · · · · · · · · · ·Xi,0 Xi,1 · · · Xi,I−i Xi,I−i+1 · · · Xi,J−1 Xi,J

· · · · · · · · · · · ·...

............

......

.........

......

· · · · · · · · · · · ·XI,0 XI,1 · · · · · · · · XI,J−1 XI,J

Denoting by Xi,j the prediction of Xi,j ∈ DcI we then have the reserve estimates :

Ri =

J∑j=I−i+1

Xi,j, i = I − J + 1, . . . , I, R =

I∑i=I−J+1

Ri .

Such predictions – including a measure of precision – can only be made based on a modelfor the random variables Xi,j; 0 ≤ i ≤ I, 0 ≤ j ≤ J.


Simplified notations

For semplicity sake we assume I = J hereafter (DI is a “triangle”).

Moreover we denote (0 ≤ i ≤ I):

· Di := Ci,I−i: cumulative claims on the last observed diagonal

· Ui := Ci,J : ultimate cost

Hence: Ri = Ui −Di

We also use the notation: x[n] :=∑n

k=0 xk


Loss Development Methods

Let us define (0 ≤ i ≤ I):

· loss development quota γi,j: percentage of Ui paid in DY j = 0, 1, . . . , J ,

· cumulative loss development quota βi,j := γi,[j].

Of course βi,J ≡ γi,[J ] ≡ 1.

Basic assumption

There exists a loss development pattern independent of the accident year :

γ := γj; 0 ≤ j ≤ J .

The cumulative quotas are given by βj := γ[j], with βJ = 1.

By the previous definitions, we can derive a number of logical relations which can be usedas the starting point for defining different stochastic models.

• E.g. Xi,j = Ui γj, 0 ≤ i, j ≤ I, suggests a cross classified model.


Moreover, let us consider the relation (0 ≤ i ≤ I):

Ui =Di

βI−i

• By adding and subtracting Di:

Ui = Di + Di1− βI−iβI−i

−→ Projective Reserve: Ri = Di1− βI−iβI−i

→ (1− βI−i)/βI−i: projection factor

• Also:

Ui = Di + Ui (1− βI−i) −→ Allocative Reserve: Ri = Ui (1− βI−i)

→ (1− βI−i): “still to come percentage”

! All these representations are equivalent in a deterministic setting, but under uncertaintythey originate a number of different stochastic models and estimation methods!


Projective (or multiplicative) methods

· Chain-Ladder (CL), and all its versions:

1− βCLjβCLj

=

J−1∏k=j

fCLk − 1 , 0 ≤ j ≤ J − 1 ,

where:

fCLj =C[I−i+1],j+1

C[I−i+1],j,

is the CL estimator of the development factor from DY j to DY j + 1.

Chain Ladder reserving is the prototype of the projective reserve.

· Models based on estimates of development factors fj := βj+1/βj different from CL.


Allocative (or additive) methods

· Bornhuetter-Ferguson (BF):

Ri = ai (1− βI−i) ,

where ai is a prior estimate of Ui.

BF reserving is the prototype of the allocative reserve

· Cape Cod (CC):

Ri = κCC Pi (1− βI−i) ,where Pi are the premiums received and κCC is an average loss ratio estimate (e.g.κCC = D[I]/

∑i PiβI−i ).

However CC can also be considered as a projection method applied to a “robusteddiagonal” DCC

i := κCC Pi βI−i.

· Additive Loss Reserving (ALR).


Under uncertainty, the fundamental question is:

What kind of information is the estimation of Ui and γj based on?

Generally:

• Projective methods only use data from “triangles“ (eventually only Di).The development pattern can also be obtained by the triangle.

? How additional information can be incorporated into the model?

• Allocative methods use prior estimates ai for the ultimate loss Ui, but need additionalinformation concerning the development pattern.

The priors ai are determined by exogenous information (external expert opinion, pric-ing, ...).

? How the “triangle” information can be incorporated?

Correct theoretical answer: Bayesian Approach

However for practical reasons “suboptimal” solutions are often used.


Hybrid models: some examples

• Naıve (classical) approach to BF model: development pattern derived by Chain-Ladder.

Widely used, but intrinsecally inconsistent: prior information not used to estimate γj.

• Hybrid Chain Ladder, by Arbenz and Salzmann (2012): weighted average of stochasticCL and BF.

• ALR model (Schmidt, 2006), BF-type model by Mack (2008). The estimate γj usesmixed information:

γj =X[I−j],j

a[I−j].

We shall denote by:

γREj =X[I−j],j

a[I−j]

(J∑k=0

X[I−k],j

a[I−k]

)−1,

the normalized version of this “raw estimator”.

• BF-type model by Saluz, Gisler and Wuthrich (2011).


• Linear hybrid model

Ri = αi Di1− βI−iβI−i

+ (1− αi) ai (1− βI−i)

How are the weights αi determined?

Ad hoc solution: αi should increase with the development of Ci,j since we obtain betterinformation on Ui with increasing time j=⇒ Benktander(1976)-Hovinen(1981): αi := βI−i.

Can we adopt a more Bayesian attitude?

Additional assumptions

In order to introduce a link between the projective and the allocative point of view weintroduce a property common to all the development years:

! The risk characteristics of accident year i are described by a latent variable Θi

! Conditionally, given Θi, the incremental losses Xi,j are independent

=⇒ Conditionally Independent Loss Increments (CILI) models.


• The approach we consider to the claims reserving problem with CILI models is basedon the linear credibility methods , which restrict the search for the best estimators tothe class of estimators which are linear functions of the observations.

• The credibility approach can be considered as an approximation of the fully Bayesianapproach. In particular cases (Exponential Dispersion Family with its associates conju-gates) credibility estimators are also exact Bayesian estimators.

(A review of fully Bayesian methods in claims reserving can be found in M.V. Wuthrich,M-Merz, 2008, sec. 4.3-4.4. See also R. Verral, 2007)

• The most representative example of a CILI model for claims reserving is theBuhlmann-Straub Credibility Reserving Model (BSCR).


The Buhlmann-Straub

Credibility Reserving Model

Basic references:

Wuthrich, M. V. and Merz, M. (2008), Stochastic Claims Reserving Methods in Insurance. Wiley Finance.

Buhlmann, H. and Moriconi, F. (2015), Credibility Claims Reserving with Stochastic Diagonal Effects. ASTIN Bulletin45(2), 309-353.

Saluz, A., Buhlmann, H., Gisler, A. and Moriconi, F. (2014), Bornhuetter-Ferguson Reserving Method with Repricing,March 17. Available at SSRN: http://ssrn.com/abstract=2697167.


Model assumptions

A1. Let Θ := Θi; 0 ≤ i ≤ I. There exist positive parameters a0, . . . , aI , γ0, . . . , γJ ,

and σ2, with∑J

j=0 γj = 1, such that for 0 ≤ i ≤ I and 0 ≤ j ≤ J :

E(Xij|Θ) = ai γj Θi ,

and:Var(Xij|Θ) = ai γj σ

2 .

A2. Let X i = Xi,j; 0 ≤ j ≤ J. For 0 ≤ i ≤ I the pairs Θi,X i are independent.Moreover, all Θ variables are independent, with:

E(Θi) = µ0 , Var(Θi) = τ 2 , 0 ≤ i ≤ I .

A1, A2 are the usual assumptions in the classical Buhlmann-Straub model, reformulatedhere in the claims reserving context:

the parameters ai; 0 ≤ i ≤ I are the given priors,

the development pattern γ = γj; 0 ≤ j ≤ J is assumed to be known.

The parameters µ0, τ2, σ2 must be estimated from the data.


The previous assumptions can be equivalently formulated in terms of Incremental LossRatios (i.e. normalized incremental losses):

Zij :=Xij

ai γj, 0 ≤ i ≤ I, 0 ≤ j ≤ J .

An estimator Θi of Θi is an estimator of E(Zi,j|Θi).

Prediction

For Xij ∈ DcI (the outstanding claims) we consider the predictor:

Xij = ai γj Θi .

Then we have the following predictor for the reserve in accident year i = 1, . . . , I and forthe total reserve:

Ri =

J∑j=I−i+1

Xij = ai (1− βI−i) Θi , R =

I∑i=1

Ri .

• We want to derive estimates for the reserve of single accident years and the totalreserve, as well as estimates of the corresponding prediction error (mean square error ofprediction, MSEP). The estimators we are looking for are in the class of the credibilityestimators.


Credible priors

If Θi is a credibility estimator, we define ai := ai Θi the credible prior of the ultimateloss Ui of accident year i. One has:

Xij = ai γj , Ri = ai (1− βI−i) .

Credibility Estimators

If the prior mean µ0 = E(Θi) is known (typically µ0 = 1), we have:

• Inhomogeneous Buhlmann-Straub estimator . The best linear inhomogeneous esti-mator for Θi (0 ≤ i ≤ I) is given by:

Θinhi = αiZ i + (1− αi)µ0 ,

where αi is the credibility weight of accident year i:

αi =ai βI−i

ai βI−i + σ2/τ 2,

and Z i is the weighted average of the incremental loss ratios observed in AY i:

Z i =

I−i∑j=0

γjβI−i

Zi,j =Di

ai βI−i.


If µ0 is not known one has:

• Homogeneous Buhlmann-Straub estimator . The best linear homogeneous estimatorfor Θi (0 ≤ i ≤ I) is given by:

Θhomi = αiZ i + (1− αi) µ0 , where µ0 =

I∑i=0

αiα[I]

Z i .

Credibility decomposition of reserve estimates

In the homogeneous case, the credible prior for accident year i is:

ahomi := ai Θhomi

= αi aiZ i + (1− αi) ai µ0

= αiDi

βI−i+ (1− αi) ai µ0 .

=⇒ credibility mixture of projective and allocative reserve:

Rhomi :=

J∑j=I−i+1

Xhomi,j = αi Di

(1− βI−i)βI−i︸︷︷︸

Projective Reserve

+(1− αi) ai µ0 (1− βI−i)︸︷︷︸Allocative Reserve

.


• If τ 2 = 0 one has αi ≡ 0 hence:

=⇒ In the inhomogeneous case (with µ0 = 1) one obtains the classsical Bornhuetter-Ferguson model:

Rinhi = ai (1− βI−i) .

=⇒ In the homogeneous case one has:

Rhomi = ai µ0 (1− βI−i) = ai κ

CC (1− βI−i) ,

where:

κCC := µ0 =

∑Ii=0Di∑I

i=0 aiβI−i,

is the Cape Cod estimate of the overall loss ratio.

Also projective point of view by defining the robusted diagonal : DCC := ai µ0 βI−i.


Numerical example

Data (Example 4.63 in Wuthrich and Merz, 2008)

A priori estimates ai of the ultimate claim and observed incremental claims Xi,j

i ai Xi,0 Xi,1 Xi,2 Xi,3 Xi,4 Xi,5 Xi,6 Xi,7 Xi,8 Xi,9

0 11,653,101 5,946,975 3,721,237 895,717 207,761 206,704 62,124 65,813 14,850 11,129 15,8141 11,367,306 6,346,756 3,246,406 723,221 151,797 67,824 36,604 52,752 11,186 11,6462 10,962,965 6,269,090 2,976,223 847,053 262,768 152,703 65,445 53,545 8,9243 10,616,762 5,863,015 2,683,224 722,532 190,653 132,975 88,341 43,3284 11,044,881 5,778,885 2,745,229 653,895 273,395 230,288 105,2245 11,480,700 6,184,793 2,828,339 572,765 244,899 104,9576 11,413,572 5,600,184 2,893,207 563,114 225,5177 11,126,527 5,288,066 2,440,103 528,0428 10,986,548 5,290,793 2,357,9369 11,618,437 5,675,568

Results

Estimated development patterns

j 0 1 2 3 4 5 6 7 8 9

fCLj 1.4925 1.0778 1.02288 1.0148 1.0070 1.0052 1 .0011 1.0011 1.0014

γCLj 0.5900 0.2904 0.0684 0.0217 0.0144 0.0069 0.0051 0.0011 0.0010 0.0014

γREj 0.5860 0.2906 0.0694 0.0224 0.0151 0.0073 0.0055 0.0012 0.0011 0.0015


Credibility estimators (d.p.: γCL)

By the usual BS estimators for the structural parameters:

τ = 0.0595, σ = 104.01929, µ0 = 0.88102 .

i αi Z i Θinhi Θhom

i ai ainhi ahomi

0 0.7924 0.9567 0.9657 0.9410 11,653,101 11,252,979 10,965,073

1 0.7880 0.9381 0.9512 0.9260 11,367,306 10,812,560 10,525,829

2 0.7817 0.9725 0.9785 0.9526 10,962,965 10,727,703 10,442,964

3 0.7760 0.9192 0.9373 0.9106 10,616,762 9,950,833 9,667,869

4 0.7819 0.8938 0.9170 0.8910 11,044,881 10,127,949 9,841,357

5 0.7873 0.8791 0.9048 0.8795 11,480,700 10,387,587 10,097,017

6 0.7838 0.8383 0.8733 0.8475 11,413,572 9,967,090 9,673,507

7 0.7756 0.7824 0.8312 0.8045 11,126,527 9,248,782 8,951,648

8 0.7600 0.7911 0.8413 0.8127 10,986,548 9,242,775 8,928,979

9 0.6917 0.8285 0.8814 0.8447 11,618,437 10,240,620 9,814,360

All Θ estimates are less then 1 ⇒ priors are pessimistic: ai < ai.

Moreover, since µ0 < µ0 = 1, homogeneous reserves will be lower than inhomogeneous.


Summary of reserve estimates with different methods

Known development pattern: γCL

i BSCRinh BSCRhom BF BH CC CL

1 15,338 14,931 16,125 15,128 14,254 15,126

2 26,419 25,718 26,999 26,259 23,866 26,257

3 35,219 34,217 37,576 34,549 33,216 34,538

4 87,511 85,035 95,434 85,389 84,361 85,302

5 161,074 156,568 178,024 156,828 157,369 156,494

6 298,051 289,272 341,306 287,771 301,705 286,121

7 477,205 461,874 574,090 455,613 507,480 449,167

8 1,109,352 1,071,689 1,318,646 1,076,297 1,165,647 1,043,242

9 4,202,908 4,027,964 4,768,385 4,286,358 4,215,123 3,950,815

Total 6,413,076 6,167,268 7,356,584 6,424,193 6,503,021 6,047,064

Total, γRE 6,573,961 6,319,544 7,505,461 6,452,322 6,644,053

(BH: Benktander-Hovinen → αi = βI−i)


Mean Square Error of Prediction of the Reserves

The expressions for the MSEPs of the single accident year reserves are derived inWuthrich and Merz (2008), Corollary 4.60.

The following expressions for the MSEPs of the total reserve are derived in Buhlmannand M. (2015), as a special case of Theorem 6.2 and Corollary 6.4:

msepR

(Rinh

)=

I∑i=1

msepRi

(Rinhi

),

msepR

(Rhom

)= msepR

(Rinh

)+τ 2

α[I]

(I∑i=1

αi(1− αi) (1− βI−i)

)2

=

I∑i=1

msepRi

(Rhomi

)+ 2

τ 2

α[I]

∑1≤i<k≤I

aiak (1− αi)(1− αk)(1− βI−i)(1− βdk) .

The last equality shows that in the homogeneous case the reserve estimates of differentaccident years are positively correlated.


Mean square errors of prediction in the BSCR model (known γ = γCL)

i Rinhi msep

1/2Ri

(Rinhi

)(%) Rhom

i msep1/2Ri

(Rhomi

)(%)

1 15,338 13,216 86.2 14,931 13,216 88.5

2 26,419 17,108 64.8 25,718 17,109 66.5

3 35,219 20,191 57.3 34,217 20,192 59.0

4 87,511 32,243 36.8 85,035 32,246 37.9

5 161,074 44,160 27.4 156,568 44,167 28.2

6 298,051 61,499 20.6 289,272 61,520 21.3

7 477,205 80,460 16.9 461,874 80,507 17.4

8 1,109,352 125,486 11.3 1,071,689 125,669 11.7

9 4,202,908 276,469 6.6 4,027,964 278,257 6.9

Total 6,413,076 326,040 5.1 6,167,268 329,031 5.3


Reserves and MSEPs in the Time Series Chain Ladder model

(Buchwalder, Buhlmann, Merz and Wuthrich, 2006)

msep1/2Ri

(Ri

)i Ri Process (%) Estimation (%) Prediction (%)

1 15,126 191 1.3 187 1.2 268 1.8

2 26,257 742 2.8 535 2.0 915 3.5

3 34,538 2,669 7.7 1,493 4.3 3,059 8.9

4 85,302 6,832 8.0 3,392 4.0 7,628 8.9

5 156,494 30,478 19.5 13,517 8.6 33,341 21.3

6 286,121 68,212 23.8 27,286 9.5 73,467 25.7

7 449,167 80,076 17.8 29,675 6.6 85,398 19.0

8 1,043,242 126,960 12.2 43,903 4.2 134,337 12.9

9 3,950,815 389,783 9.9 129,770 3.3 410,818 10.4

Total 6,047,064 424,380 7.0 185,026 3.1 462,961 7.7


MSEP under the one-year view

For solvency purposes the MSEPs should be computed under a “one-year view”, i.e. therandom variable to be considered should be the Year-End Obligations (next-diagonalpayments plus residual reserve), or equivalently the one-year Claims Development Result(CDR).

In credibility-based claims reserving models, closed form expressions for MSEPs underthe one-year view have been derived:

· in Buhlmann, De Felice, Gisler, M. and Wuthrich (2008) for the Credibility ChainLadder model,

· in Merz and Wuthrich (2011) for the Credibility-Based Additive Loss Leservingmodel.

A similar approach can be used in the BSCR model. The basic idea is to express the year-end estimate αI+1

i of the credibility weight as a linear updating of the current estimateαIi .


BSCR model with unknown development pattern

(Work in progress, jointly with Hans Buhlmann and Mario Wuthrich)

If we relax the assumption of known development pattern, the quotas γ must be estimatedfrom the data.

Estimation of the development pattern

We propose an iterative procedure. The basic idea is that if the parameters Θ would beknown, then the best linear unbiased estimator of γj would be:

γrawj :=

∑I−ji=0 Xi,j∑I−ji=0 ai Θi

, j = 0, . . . , J .

Since the Θ variables are unknown they are replaced by some estimates. Moreover, somekind of normalization is needed.For the inhomogeneous/homogeneous case we choose the following estimator:

γinh/homj =

∑I−ji=0 Xi,j∑I−j

i=0 ai Θinh/homi

(J∑l=0

∑I−li=0Xi,l∑I−l

i=0 ai Θinh/homi

)−1, j = 0, . . . , J . (1)


a) The estimated pattern γ will depend on Θ, that is γ = γ(Θ);

b) in turn, Θ depends itself by γ, that is Θ = Θ(γ).

Definition. We call (Θ, γ) a compatible pair if Θ(γ) = Θ and γ(Θ) = γ

=⇒Iterative estimation algorithm (homogeneous case):

0. Initialization. We choose an initial development pattern γ(0) and pose γ = γ(0).

1. Estimation of the structural parameters. Using γ, estimates of the parameters µ0, τ2, σ2 are

obtained by the data.

2. Estimation of Θ. Using γ and the parameter estimates obtained in step 1, the estimates Θ are

derived.

3. Estimation of γ. Using Θ, a new d.p. estimate γ ′ is computed.

4. Iteration. If a given convergence criterion is not fulfilled, we return to step 1 assuming for γ the

development pattern γ ′. Otherwise the iteration is stopped and we adopt the resulting estimates Θ

and γ as the compatible pair (Θ, γ).

convercence criterion: max

√√√√ J∑

j=0

(γ(n)j − γ

(n−1)j

)2,

√√√√ I∑i=0

(Θ

(n)i − Θ

(n−1)i

)2 < ε ,


Reserve estimates and MSEPs with unknown γ

Inhomogeneous case. After 5 iterations (with ε = 10−7) we obtained:

τ = 0.05926 , σ = 103.76437 .

Homogeneous case. After 4 iterations:

τ = 0.05926 , σ = 103.76583 , µ0 = 0.88133 .

i αi Rinhi msep

1/2Ri

(%) Rhomi msep

1/2Ri

(%)

0 0.7917 0 0 . 0 0 .1 0.7873 15,596 13,294 85.2 15,181 13,294 87.62 0.7810 26,844 17,203 64.1 26,128 17,202 65.83 0.7753 35,797 20,306 56.7 34,775 20,306 58.44 0.7812 88,896 32,416 36.5 86,373 32,419 37.55 0.7866 163,437 44,372 27.1 158,852 44,378 27.96 0.7831 301,931 61,742 20.4 293,014 61,762 21.17 0.7747 482,521 80,699 16.7 466,986 80,746 17.38 0.7590 1,117,632 125,627 11.2 1,079,625 125,811 11.79 0.6904 4,217,905 276,202 6.5 4,042,181 278,000 6.9

Total 6,450,559 326,035 5.1 6,203,114 329,048 5.3

Total, γCL 6,413,076 326,040 5.1 6,167,268 329,031 5.3


Deriving the estimation error

If the develpment pattern is unknown, we need also an assessment of the prediction erroron the reserve estimates due to the estimation error of γ.

We define this estimation error for the reserve estimate for accident year i = 1, . . . , I as:

EE(γ)i := E

(R

(γ)i − Ri

)2,

· R(γ)i is the reserve estimate obtained with the “true value” of γ,

· Ri is the reserve estimate derived by the iterative algorithm.

For the total reserve we pose:

EE(γ) :=

(I∑i=1

E(R

(γ)i − Ri

))2

.

In order to derive an estimate of EE(γ)i and EE(γ) we follow a parametric bootstrap

simulation method partially similar to that used in Saluz, Buhlmann, Gisler and M.(2014).


The bootstrap simulation procedure

In the bootstrap simulation a large number of “pseudo-triangles” DI is simulated and foreach pseudo-triangle two reserve estimates are computed for each one of these pseudo-data:

• one estimate is obtained using as development pattern the estimate γ resulting bythe original iterative procedure (which is assumed to provide the “true developmentpattern”),

• the other estimate is obtained by a new iterative estimation procedure (thereforewe run the estimation algorithm in each simulation), without re-estimation of thestructutal parameters.

An assessment of the estimation error is then obtained as the average of the squaredifferences of the two reserve estimate.

Remark. Differently from the approach used in Saluz, Buhlmann, Gisler and M. (2014),in the bootstrap simulations we use the original estimates for the structural parameters inall the iterations in order to avoid to include the estimation error of these hyperparametersinto the estimation error of γ.


Let us denote by τ 2, σ2, µ0 the parameter estimates obtained for the compatible pair (Θ, γ).

In the s-th simulation (s = 1, . . . , S) we have the following steps:

1. The random variables Θ, ε are generated as independent and normally distributed, with:

Θ(s)i ∼ N(µ0, τ

2) , ε(s)i,j ∼ N(1, 0) , 0 ≤ i + j ≤ I .

2. A pseudo-trapezoid of incremental claims is generated as:

X(s)i,j = ai γj Θ

(s)i +

√ai γj σ ε

(s)i,j , 0 ≤ i + j ≤ I .

3. The reserve estimates R(s)i and R(s) =

∑i R

(s)i , are obtained by this data assuming the original

development pattern γ and using the original parameter estimates τ 2 , σ2, µ0.

4. The previous iterative estimation procedure is applied with γ(0) = γ and where step 1 is skipped,

so that the original parameter estimates τ 2 , σ2, µ0 are used in each iteration. The simulated reserve

estimates R(s)i and R(s) =

∑i R

(s)i are then derived by the compatible pair (Θ

(s), γ(s)) provided by

the iterations.

5. The squared errors are computed:

∆(s)i =

(R

(s)i − R

(s)i

)2, i = 1, . . . , I, ∆(s) =

(I∑i=1

(R

(s)i − R

(s)i

))2

.

The estimation errors are estimated as EE(γ)

i = 1S

∑Ss=1 ∆

(s)i and EE

(γ)= 1

S

∑Ss=1 ∆(s).


MSEP in the BSCR model including γ estimation error

S = 10000, 3 ≤ num iterations ≤ 5

i msep1/2Ri

(Rinhi

)(%) msep

1/2Ri

(Rhomi

)(%)

1 19,072 122.3 18,783 123.7

2 23,140 86.2 22,829 87.4

3 26,067 72.8 25,754 74.1

4 38,856 43.7 38,516 44.6

5 51,524 31.5 51,166 32.2

6 69,385 23.0 69,025 23.6

7 88,730 18.4 88,386 18.9

8 135,231 12.1 134,973 12.5

9 291,912 6.9 292,987 7.2

Total 395,536 6.1 395,910 6.4

Total no EE 326,035 5.1 329,048 5.3


The convergence issue

In all the numerical examples we considered, the iterative estimation algorithm everconverged to a unique point (Θ, γ), independently of the initial d.p. γ(0).

However the convergence of the algorithm should be studied (and possibly proven) usingproper theoretical arguments. Technically, one should prove that the algorithm is acontraction.

We studied the case without re-estimation of the structural parameters.

A proof of convergence is available (provided by M. Wuthrich) for the case I = J = 1(the 1-dimensional case, since γ0 + γ1 = 1).

For I, J > 1 the functions Θ(γ) and γ(Θ) are difficult to be studied analitically and, atthe moment, we can oly conjecture that the convergence to a unique pair holds in thegeneral case.


Adding stochastic diagonal effects:

the BSCR Model with Additive Diagonal Risk

(ADR Model)

Basic references:

Buhlmann, H. and Moriconi, F. (2015), Credibility Claims Reserving with Stochastic Diagonal Effects. ASTIN Bulletin45(2), 309-353.


Model assumptions

A1. Let Θ := ηi, ζi+j; 0 ≤ i ≤ I, 0 ≤ j ≤ J. There exist positive parameters

a0, . . . , aI , γ0, . . . , γJ , and σ2, with∑J

j=0 γj = 1, such that for 0 ≤ i ≤ I and 0 ≤ j ≤ J :

E(Xij|Θ) = ai γj (ηi + ζi+j)

and:Var(Xij|Θ) = ai γj σ

2 .

A2. All η, ζ variables are independent, with:

E(ηi) = µ0 , Var(ηi) = τ 2 , 0 ≤ i ≤ I ,E(ζi+j) = 0 , Var(ζi+j) = χ2 , 0 ≤ i ≤ I , 0 ≤ j ≤ J .

We interpret:

· ηi: random effect of accident year i,

· ζi+j: random effect of calendar year t = i + j (random diagonal effect).

As usual the parameters ai are the given priors, the development quotas γ are knownand the parameters µ0, τ

2, χ2, σ2 must be estimated from the data.


Innovative aspects of the ADR model:

• A calendar year effect is included in the latent variables Θ which takes the form:

Θi,t = ηi + ζt for the calendar year t = i + j ,

thus the calendar year effect is separated additively from the (random) accidentyear effect.

• Independence between accident years is relaxed. Our assumptions imply thefollowing correlation structure for the risk parameters:

Cov(Θit,Θks) = τ 2 IIi,k + χ2 IIt,s , 0 ≤ i, k ≤ I , 0 ≤ t, s ≤ I + J ,

with IIi,k = 1 if i = k and 0 elsewhere.The correlation within the same accident year i is induced by ηi and is proportionalto τ 2; the correlation within the same calendar year t is induced by ζt and isproportional to χ2.

! If χ2 = 0 the ADR model reduces to the BSCR model (with Θi ≡ ηi).


• Using a full Bayesian approach, diagonal risk effects are also treated by Shi, Basu andMeyer (2012) and by Wuthrich (2012, 2013).

Remark. The ADR assumptions implicitly provide an extension of the classical Buhlmann-Straub model that can be used also in applications different from claims reserving.

For example, for the classical applications in premium rating :

– rewrite the assumptions for Zi,t := Xi,t/aiγt−i,

– interpret Zi,t as the loss ratio of company i (risk i in a collective) observed in year t,

– use wi,t := aiγt−i as the associated weight.


Prediction

For Xij ∈ DcI , we now consider the predictor Xij = ai γj ηi.

Again, the estimators we are looking for are in the class of the credibility estimators.

Credible priors: ai := ai ηi =⇒ Xij = ai γj , Ri = ai (1− βI−i).However, since the accident years are no more independent, credibility formulae are morecomplex than in the classical case.

Covariance matrix of the observations

For any pair of observations Xi,j, Xk,l ∈ DI , we consider the covariance between theincremental loss ratios:

ω(i,j),(k,l) := Cov(Zi,j, Zk,l) .

With this double-index notation, the covariance matrix of the observations is:

Ω :=(ω(i,j),(k,l)

)(i,j),(k,l)∈DI

.

To fix ideas, we choose for Ω the left-to-right/top-to-bottom ordering.

Under the model assumptions one has:

Cov(Zij, Zkl) = τ 2 IIi,k + χ2 IIi+j,k+l +σ2

ai γjIIi,k IIj,l , 0 ≤ i, k ≤ I , 0 ≤ j, l ≤ J .


Structure of the Ω matrix with I = J = 2

ν 1 2 3 4 5 6

λ (0,0) (0,1) (0,2) (1,0) (1,1) (2,0)

1 (0,0) τ 2 + χ2 + σ2

a0γ0τ 2 τ 2 0 0 0

2 (0,1) τ 2 τ 2 + χ2 + σ2

a0γ1τ 2 χ2 0 0

3 (0,2) τ 2 τ 2 τ 2 + χ2 + σ2

a0γ20 χ2 χ2

4 (1,0) 0 χ2 0 τ 2 + χ2 + σ2

a1γ0τ 2 0

5 (1,1) 0 0 χ2 τ 2 τ 2 + χ2 + σ2

a1γ1χ2

6 (2,0) 0 0 χ2 0 χ2 τ 2 + χ2 + σ2

a2γ0

• If χ2 = 0 (BSCR model) the covariance matrix Ω is block diagonal , where eachblock corresponds to an accident year.If Ω is a block diagonal matrix with blocks Bi, the inverse matrix Ω−1 is also blockdiagonal, with blocks B−1i .By this property, in the classical credibility theory (hence in the BSCR) all therelevant expressions of accident year i (e.g. expressions for αi, Z i and ηi) are basedonly on the observations of accident year i.

• If χ2 > 0 (ADR model) Ω is not block diagonal, hence the relevant expressions foraccident year i depend now on all the observations.


Theorem 4.1. The best linear inhomogeneous estimator for ηi, 0 ≤ i ≤ I, is givenby:

ηinhi = αi¯Zi + µ0 (1− αi) ,

where:

αi := τ 2I∑

k=0

I−k∑l=0

I−i∑j=0

ω(−1)(i,j),(k,l) ,

¯Zi :=τ 2

αi

I∑k=0

I−k∑l=0

I−i∑j=0

ω(−1)(i,j),(k,l)Zk,l .

The Ω−1 matrix with I = J = 2

ν 1 2 3 4 5 6

λ (0,0) (0,1) (0,2) (1,0) (1,1) (2,0)

1 (0,0) ω−1(0,0),(1,0) ω−1(0,0),(1,1) ω−1(0,0),(2,0)

2 (0,1) B−10 ω−1(0,1),(1,0) ω−1(0,1),(1,1) ω−1(0,1),(2,0)

3 (0,2) ω−1(0,2),(1,0) ω−1(0,2),(1,1) ω−1(0,2),(2,0)

4 (1,0) ω−1(1,0),(0,0) ω−1(1,0),(0,1) ω−1(1,0),(0,2) B−11 ω−1(1,0),(2,0)

5 (1,1) ω−1(1,1),(0,0) ω−1(1,1),(0,1) ω−1(1,1),(0,2) ω−1(1,1),(2,0)

6 (2,0) ω−1(2,0),(0,0) ω−1(2,0),(0,1) ω−1(2,0),(0,2) ω−1(2,0),(1,0) ω−1(2,0),(1,1) B−12

αi is obtained by Ω−1 by computing the sum of all the elements on the band correspondingto accident year i (and then multiplying by τ 2).


Theorem 4.4. The best linear homogeneous estimator for ηi, 0 ≤ i ≤ I, is givenby:

ηhomi = αi¯Zi + (1− αi) µ0 , with µ0 =

I∑i=0

αiα•

¯Zi .

Credibility decomposition of reserve estimates

Rhomi :=

J∑j=I−i+1

Xhomi,j = αi ai

¯Zi (1− βI−i)︸︷︷︸Projective Reserve

+(1− αi) ai µ0 (1− βI−i)︸︷︷︸Allocative Reserve

For τ 2 = 0 we again obtain Bornhuetter-Ferguson model and Cape Cod model as specialcases.

Mean Square Error of Prediction of the Reserves

Closed form expressions are obtained both in the inhomogeneous and the homogeneouscase.


Parameter estimation

Estimates of τ 2, χ2, σ2 are obtained as the solution of a system of 3 linear equationsinvolving sums of square errors.

A method typical in the analysis of variance:

Some types of sum of square errors (SS) are taken from the data and the expecta-tion E(SS) of each of these sums is expressed as a function f (τ 2, χ2, σ2). If the ex-pectation E(SS) is replaced by the corresponding observed value SS∗, the equationsSS∗ = f (τ 2, χ2, σ2) can provide sufficient constraints to identify (i.e. estimate) thevariance parameters.

We apply this method considering three different SS taken on the incremental loss ratiosin the observed trapezoid DI . Given model assumptions, the f functions are linear, andwe are led to a system of three independent linear equations. We obtain estimates forτ 2, χ2, σ2 by solving this system.

Because of the linearity of the equations these estimates are unbiased.

If χ2 = 0 this method provides the same estimators of the classical BS model.


Reserves and MSEP in the ADR model (known γ = γCL)

Structural parameters. τ = 0.04961, χ = 0.05755, σ = 63.233, µ0 = 0.88204.

Due to the additional uncertainty, the αi in ADR are substantially smaller than in BSCR

=⇒ ADR reserves are closer to the BF-type reserves

i αi Rinhi msep

1/2Ri

(Rinhi

)(%) Rhom

i msep1/2Ri

(Rhomi

)(%)

0 0.4405 0 0 . 0 0 .1 0.4090 15,155 10,620 70.1 14,031 10,623 75.72 0.3952 26,683 13,743 51.5 24,757 13,749 55.53 0.3867 36,544 16,219 44.4 33,825 16,229 48.04 0.3848 91,926 26,089 28.4 85,000 26,132 30.75 0.3829 170,354 35,945 21.1 157,395 36,054 22.96 0.3769 320,635 50,801 15.8 295,551 51,088 17.37 0.3668 511,867 67,539 13.2 468,989 68,170 14.58 0.3487 1,208,764 113,536 9.4 1,107,452 115,623 10.49 0.3047 4,620,160 316,789 6.9 4,229,107 327,843 7.8

Total 7,002,087 407,426 5.8 6,416,109 426,609 6.6

Total in BSCR 6,413,077 326,040 5.1 6,167,268 329,031 5.3


ADR model with unknown development pattern

(Work in progress, jointly with Hans Buhlmann and Mario Wuthrich)

The approach we are following is essentially the same we applied to BSCR model.

Assuming that an estimate ηinh/homi has been obtained by the previous formulae, we

adopt for γj the normalized estimator:

γinh/homj =

∑I−ji=0 Xi,j∑I−j

i=0 ai ηinh/homi

(J∑l=0

∑I−li=0Xi,l∑I−l

i=0 ai ηinh/homi

)−1, j = 0, . . . , J .

Using these estimators, we can introduce the definition of compatible pair (η, γ) andspecify an iterative estimation algorithm essentially similar to that used in the BSCRmodel.

Estimation error of development pattern

A parametric bootstrap simulation method is used essentially similar to that used forBSCR.


Reserves and MSEP in the ADR model including γ estimation error

S = 10000, 4 ≤ num iterations ≤ 7

i Rinhi msep

1/2Ri

(Rinhi

)(%) Rhom

i msep1/2Ri

(Rhomi

)(%)

1 15,315 15,401 100.6 14,146 14,744 104.2

2 27,383 18,740 68.4 25,354 18,040 71.2

3 37,840 21,127 55.8 34,957 20,437 58.5

4 95,862 31,706 33.1 88,478 30,967 35.0

5 177,582 42,348 23.8 163,790 41,612 25.4

6 333,265 57,968 17.4 306,705 57,409 18.7

7 529,862 75,392 14.2 484,771 75,235 15.5

8 1,241,745 123,760 10.0 1,136,370 125,278 11.0

9 4,701,896 335,611 7.1 4,301,428 347,447 8.1

Total 7,160,751 471,848 6.6 6,555,999 488,943 7.5

Total no EE 7,002,087 411,900 5.8 6,416,109 432,646 6.6


The convergence issue

As concerning the convergence of the estimation algorithm, however, the situation isdiffent from the BSCR case.

• Counterexamples, i.e. examples of not convergence, can be found also for the casewithout re-estimation of the structural parameters.

• However this happens in rather extreme/unrealistic cases and it seems that we haveconvergence in most practical situations.? convergence fails for very large values of the ratio χ2/σ2?


franco moriconi - actuaries.chb9f3124e...bf-type model by saluz, gisler and wuthric h (2011). saa...

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